Machine learning comes in two varieties:
\[\begin{equation} C(f(X_i|\theta), \;y_{i}^\text{actual}) \geq 0. \end{equation}\]
\[\begin{equation} \theta = \underset{\theta}{\operatorname{argmin}} \sum C(f(X_i|\theta), \;y_{i}^\text{actual}) + \lambda \left\lVert\theta\right\rVert, \end{equation}\]
where \(\lambda\) is some penalty for overfitting (more tomorrow)
\(\implies\) use language of probability
\[\begin{equation} p(X|\text{camera quality, location, cat genetics etc.)}. \end{equation}\]
\[\begin{equation} p(X, \text{camera quality, location, cat genetics etc.}) \end{equation}\]
\[\begin{equation} \widehat p_\theta(y|Z) \approx p(y|X), \end{equation}\]
where \(Z\) are features of \(X\)
\[\begin{equation} \widehat p_\theta(Z) \approx p(X), \end{equation}\]
where \(Z\) are features of \(X\)
| country | designation | points | price | province | region_1 | region_2 | variety | winery |
|---|---|---|---|---|---|---|---|---|
| US | Martha’s Vineyard | 96 | 235 | California | Napa Valley | Napa | Cabernet Sauvignon | Heitz |
| Spain | Carodorum Selección Especial Reserva | 96 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | |
| US | Special Selected Late Harvest | 96 | 90 | California | Knights Valley | Sonoma | Sauvignon Blanc | Macauley |
| US | Reserve | 96 | 65 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi |
| France | La Brûlade | 95 | 66 | Provence | Bandol | Provence red blend | Domaine de la Bégude | |
| Spain | Numanthia | 95 | 73 | Northern Spain | Toro | Tinta de Toro | Numanthia | |
| Spain | San Román | 95 | 65 | Northern Spain | Toro | Tinta de Toro | Maurodos | |
| Spain | Carodorum Único Crianza | 95 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | |
| US | Silice | 95 | 65 | Oregon | Chehalem Mountains | Willamette Valley | Pinot Noir | Bergström |
| US | Gap’s Crown Vineyard | 95 | 60 | California | Sonoma Coast | Sonoma | Pinot Noir | Blue Farm |
| Italy | Ronco della Chiesa | 95 | 80 | Northeastern Italy | Collio | Friulano | Borgo del Tiglio | |
| US | Estate Vineyard Wadensvil Block | 95 | 48 | Oregon | Ribbon Ridge | Willamette Valley | Pinot Noir | Patricia Green Cellars |
| US | Weber Vineyard | 95 | 48 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Patricia Green Cellars |
| France | Château Montus Prestige | 95 | 90 | Southwest France | Madiran | Tannat | Vignobles Brumont | |
| US | Grace Vineyard | 95 | 185 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Domaine Serene |
| US | Sigrid | 95 | 90 | Oregon | Willamette Valley | Willamette Valley | Chardonnay | Bergström |
| US | Rainin Vineyard | 95 | 325 | California | Diamond Mountain District | Napa | Cabernet Sauvignon | Hall |
| Spain | 6 Años Reserva Premium | 95 | 80 | Northern Spain | Ribera del Duero | Tempranillo | Valduero | |
| France | Le Pigeonnier | 95 | 290 | Southwest France | Cahors | Malbec | Château Lagrézette | |
| US | Gap’s Crown Vineyard | 95 | 75 | California | Sonoma Coast | Sonoma | Pinot Noir | Gary Farrell |
| US | Grignolino | 95 | 24 | California | Napa Valley | Napa | Rosé | Heitz |
| Spain | Prado Enea Gran Reserva | 95 | 79 | Northern Spain | Rioja | Tempranillo Blend | Muga | |
| Spain | Termanthia | 95 | 220 | Northern Spain | Toro | Tinta de Toro | Numanthia | |
| US | Giallo Solare | 95 | 60 | California | Edna Valley | Central Coast | Chardonnay | Center of Effort |
| US | R-Bar-R Ranch | 95 | 45 | California | Santa Cruz Mountains | Central Coast | Pinot Noir | Comartin |
| New Zealand | Maté’s Vineyard | 94 | 57 | Kumeu | Chardonnay | Kumeu River | ||
| US | Shea Vineyard | 94 | 62 | Oregon | Willamette Valley | Pinot Noir | Bergström | |
| US | Abetina | 94 | 105 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi |
| US | Garys’ Vineyard | 94 | 60 | California | Santa Lucia Highlands | Central Coast | Pinot Noir | Roar |
| US | The Funk Estate | 94 | 60 | Washington | Walla Walla Valley (WA) | Columbia Valley | Syrah | Saviah |
| Bulgaria | Bergulé | 90 | 15 | Bulgaria | Mavrud | Villa Melnik | ||
| US | Babushka | 90 | 37 | California | Russian River Valley | Sonoma | Chardonnay | Zepaltas |
| Italy | Vigna Piaggia | 90 | NA | Tuscany | Brunello di Montalcino | Sangiovese | Abbadia Ardenga | |
| France | Nonpareil Trésor Rosé Brut | 90 | 22 | France Other | Vin Mousseux | Sparkling Blend | Bouvet-Ladubay | |
| US | Conner Lee Vineyard | 90 | 42 | Washington | Columbia Valley (WA) | Columbia Valley | Chardonnay | Buty |
| Italy | Riserva | 90 | 135 | Tuscany | Brunello di Montalcino | Sangiovese | Carillon | |
| France | 90 | 60 | Rhône Valley | Châteauneuf-du-Pape | Rhône-style White Blend | Clos de L’Oratoire des Papes | ||
| Italy | 90 | 29 | Tuscany | Vino Nobile di Montepulciano | Sangiovese | Avignonesi | ||
| Italy | 90 | 23 | Tuscany | Chianti Classico | Sangiovese | Casina di Cornia | ||
| Italy | Riserva | 90 | 29 | Tuscany | Chianti Classico | Red Blend | Castello di Monterinaldi | |
| Spain | Amandi | 90 | 17 | Galicia | Ribeira Sacra | Mencía | Don Bernardino | |
| Spain | Alfonso Oloroso Seco | 90 | 26 | Andalucia | Jerez | Palomino | González Byass | |
| US | Private Reserve | 90 | 55 | Idaho | Petite Sirah | Huston | ||
| Italy | Riserva | 90 | 39 | Tuscany | Chianti Classico | Red Blend | Rignana | |
| France | Coteaux | 90 | 69 | Rhône Valley | Cornas | Syrah | Tardieu-Laurent | |
| Italy | Vigneto Odoardo Beccari Riserva | 90 | 30 | Tuscany | Chianti Classico | Red Blend | Vignavecchia | |
| Italy | Poggio alle Mura | 90 | 90 | Tuscany | Brunello di Montalcino | Sangiovese | Banfi | |
| US | Estate Grown | 90 | 60 | California | Mount Veeder | Napa | Cabernet Sauvignon | Brandlin |
| Italy | 90 | 50 | Tuscany | Brunello di Montalcino | Sangiovese | Brunelli Martoccia | ||
| US | 90 | 40 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | Canvasback | |
| Italy | Riserva | 90 | 100 | Tuscany | Brunello di Montalcino | Sangiovese | Capanne Ricci | |
| France | 90 | 68 | Burgundy | Chassagne-Montrachet | Chardonnay | Chartron et Trébuchet | ||
| France | Les 7 Hommes | 90 | 42 | Loire Valley | Sancerre | Sauvignon Blanc | Cherrier Frères | |
| France | L’Inédit | 90 | 28 | Loire Valley | Coteaux du Giennois | Pinot Noir | Clement et Florian Berthier | |
| US | 90 | 18 | California | Russian River Valley | Sonoma | Chardonnay | De Loach | |
| US | Four Flags | 90 | 69 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | DeLille |
| France | Le Pavé | 90 | NA | Loire Valley | Sancerre | Sauvignon Blanc | Domaine Vacheron | |
| US | Reserve | 90 | 25 | New York | Finger Lakes | Finger Lakes | Riesling | Dr. Konstantin Frank |
| US | Final Final | 90 | 30 | Washington | Columbia Valley (WA) | Columbia Valley | Cabernet Sauvignon-Syrah | Efeste |
| Italy | Poggio Bestiale | 90 | 60 | Tuscany | Maremma Toscana | Red Blend | Fattoria di Magliano | |
| Argentina | The Apple Doesn’t Fall Far From The Tree | 91 | 30 | Mendoza Province | Mendoza | Malbec | Matias Riccitelli | |
| Australia | 91 | 36 | Victoria | Mornington Peninsula | Pinot Noir | Moorooduc | ||
| Argentina | Alegoría Gran Reserva | 91 | 25 | Mendoza Province | Mendoza | Malbec | Navarro Correas | |
| France | L’Homme Mort Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Chenevières | |
| Portugal | 91 | 23 | Alentejano | Portuguese Red | Herdade do Rocim | |||
| US | Estate Select | 91 | 36 | California | Santa Clara Valley | Central Coast | Syrah | Jason-Stephens |
| France | Fourchaume Premier Cru | 91 | 38 | Burgundy | Chablis | Chardonnay | Louis Max | |
| US | Animo | 91 | 85 | California | Napa Valley | Napa | Cabernet Sauvignon | Michael Mondavi Family Estate |
| US | Schindler Vineyard | 91 | 50 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Panther Creek |
| US | Barrel Select | 91 | 60 | California | Rutherford | Napa | Cabernet Sauvignon | Provenance Vineyards |
| US | District Collection | 91 | 85 | California | St. Helena | Napa | Cabernet Sauvignon | Raymond |
| US | 91 | 45 | California | Sonoma Coast | Sonoma | Pinot Noir | Red Car | |
| Italy | Bussia Riserva | 91 | NA | Piedmont | Barolo | Nebbiolo | Silvano Bolmida | |
| US | 91 | 19 | Oregon | Willamette Valley | Willamette Valley | Pinot Gris | Trinity Vineyards | |
| Portugal | Premium | 91 | 15 | Alentejo | Portuguese Red | Adega Cooperativa de Borba | ||
| US | Premier Cuvée | 91 | 54 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Archery Summit |
| France | Le Nombre d’Or Brut Nature | 91 | 85 | Champagne | Champagne | Chardonnay | Aubry | |
| US | Juliana Vineyard | 91 | 38 | California | Napa Valley | Napa | Sauvignon Blanc | B Cellars |
| US | 91 | 28 | California | Napa Valley | Napa | Cabernet Sauvignon | B Side | |
| Italy | Boscato | 91 | 75 | Piedmont | Barolo | Nebbiolo | Bel Colle | |
| US | Aeolian | 91 | 42 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Bethel Heights |
| Israel | Reserve | 91 | 25 | Upper Galilee | Cabernet Sauvignon | Binyamina | ||
| Italy | Palliano Riserva | 91 | NA | Piedmont | Roero | Nebbiolo | Ceste | |
| Italy | del Comune di Serralunga d’Alba | 91 | 59 | Piedmont | Barolo | Nebbiolo | Cascina Cucco | |
| Italy | Bricco Luciani | 91 | 85 | Piedmont | Barolo | Nebbiolo | Cascina del Monastero | |
| Italy | Bricco Gattera | 91 | 80 | Piedmont | Barolo | Nebbiolo | Cordero di Montezemolo | |
| France | Montmains Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Gérard Duplessis | |
| US | 91 | 22 | California | Napa Valley | Napa | Cabernet Sauvignon | Eagle Glen | |
| US | Bacigalupi Vineyard | 91 | 65 | California | Russian River Valley | Sonoma | Pinot Noir | Eleven Eleven |
| US | Magnificat | 91 | 50 | California | Napa Valley | Napa | Meritage | Franciscan |
| US | 86 | 10 | California | California | California Other | Cabernet Sauvignon | Belle Ambiance | |
| Portugal | Marquês de Marialva Rosé Bruto | 86 | 12 | Beira Atlantico | Baga | Adega de Cantanhede | ||
| Italy | Nature | 86 | 22 | Veneto | Prosecco | Glera | De Stefani | |
| US | Small Lot Blend | 86 | 13 | California | Mendocino County | Mendocino/Lake Counties | Chardonnay | Parducci |
| Portugal | Muros de Vinha | 86 | 10 | Douro | Portuguese Red | Quinta do Portal | ||
| France | Château Beauvillain-Monpezat | 86 | 14 | Southwest France | Cahors | Malbec-Merlot | Rigal | |
| US | 86 | 18 | California | California | California Other | Chardonnay | The Naked Grape | |
| US | 86 | 36 | California | Sonoma Valley | Sonoma | Cabernet Sauvignon | Tin Barn | |
| France | Pigmentum | 86 | 15 | Southwest France | Buzet | Merlot-Malbec | Georges Vigouroux | |
| France | Pigmentum | 86 | 10 | Southwest France | Côtes de Gascogne | Ugni Blanc-Colombard | Georges Vigouroux |
\[\begin{equation} (x_1,x_2)' \sim \mathcal{N}(\mu, \Sigma) \end{equation}\]
where \(\Sigma\) is dense.
\[\begin{equation} (y_1,y_2)' \sim \mathcal{N}(0, D) \end{equation}\]
where \(D\) is diagonal.
Remember, we’ve assumed:
\[\begin{equation} (x_1,x_2)' \sim \mathcal{N}(\mu, \Sigma) \end{equation}\]
\[\begin{equation} (\tilde x_{1,i}, \tilde x_{2,i}) = (x_{1,i}, x_{2,i}) - (\bar x_{1}, \bar x_{2}) \end{equation}\]
\[\begin{equation} \widehat{\Sigma} = \frac{1}{n}\sum (\tilde x_{1,i}, \tilde x_{2,i})' (\tilde x_{1,i}, \tilde x_{2,i}) \end{equation}\]
\[\begin{equation} \widehat{\Sigma} = P D P' \end{equation}\]
\[\begin{align} y_1 &= P_1' . (\tilde x_1, \tilde x_2)\\ y_2 &= P_2' . (\tilde x_1, \tilde x_2) \end{align}\]
| country | designation | points | price | province | region_1 | region_2 | variety | winery | review_id |
|---|---|---|---|---|---|---|---|---|---|
| US | Martha’s Vineyard | 96 | 235 | California | Napa Valley | Napa | Cabernet Sauvignon | Heitz | 1 |
| Spain | Carodorum Selección Especial Reserva | 96 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | 2 | |
| US | Special Selected Late Harvest | 96 | 90 | California | Knights Valley | Sonoma | Sauvignon Blanc | Macauley | 3 |
| US | Reserve | 96 | 65 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi | 4 |
| France | La Brûlade | 95 | 66 | Provence | Bandol | Provence red blend | Domaine de la Bégude | 5 | |
| Spain | Numanthia | 95 | 73 | Northern Spain | Toro | Tinta de Toro | Numanthia | 6 | |
| Spain | San Román | 95 | 65 | Northern Spain | Toro | Tinta de Toro | Maurodos | 7 | |
| Spain | Carodorum Único Crianza | 95 | 110 | Northern Spain | Toro | Tinta de Toro | Bodega Carmen Rodríguez | 8 | |
| US | Silice | 95 | 65 | Oregon | Chehalem Mountains | Willamette Valley | Pinot Noir | Bergström | 9 |
| US | Gap’s Crown Vineyard | 95 | 60 | California | Sonoma Coast | Sonoma | Pinot Noir | Blue Farm | 10 |
| Italy | Ronco della Chiesa | 95 | 80 | Northeastern Italy | Collio | Friulano | Borgo del Tiglio | 11 | |
| US | Estate Vineyard Wadensvil Block | 95 | 48 | Oregon | Ribbon Ridge | Willamette Valley | Pinot Noir | Patricia Green Cellars | 12 |
| US | Weber Vineyard | 95 | 48 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Patricia Green Cellars | 13 |
| France | Château Montus Prestige | 95 | 90 | Southwest France | Madiran | Tannat | Vignobles Brumont | 14 | |
| US | Grace Vineyard | 95 | 185 | Oregon | Dundee Hills | Willamette Valley | Pinot Noir | Domaine Serene | 15 |
| US | Sigrid | 95 | 90 | Oregon | Willamette Valley | Willamette Valley | Chardonnay | Bergström | 16 |
| US | Rainin Vineyard | 95 | 325 | California | Diamond Mountain District | Napa | Cabernet Sauvignon | Hall | 17 |
| Spain | 6 Años Reserva Premium | 95 | 80 | Northern Spain | Ribera del Duero | Tempranillo | Valduero | 18 | |
| France | Le Pigeonnier | 95 | 290 | Southwest France | Cahors | Malbec | Château Lagrézette | 19 | |
| US | Gap’s Crown Vineyard | 95 | 75 | California | Sonoma Coast | Sonoma | Pinot Noir | Gary Farrell | 20 |
| US | Grignolino | 95 | 24 | California | Napa Valley | Napa | Rosé | Heitz | 21 |
| Spain | Prado Enea Gran Reserva | 95 | 79 | Northern Spain | Rioja | Tempranillo Blend | Muga | 22 | |
| Spain | Termanthia | 95 | 220 | Northern Spain | Toro | Tinta de Toro | Numanthia | 23 | |
| US | Giallo Solare | 95 | 60 | California | Edna Valley | Central Coast | Chardonnay | Center of Effort | 24 |
| US | R-Bar-R Ranch | 95 | 45 | California | Santa Cruz Mountains | Central Coast | Pinot Noir | Comartin | 25 |
| New Zealand | Maté’s Vineyard | 94 | 57 | Kumeu | Chardonnay | Kumeu River | 26 | ||
| US | Shea Vineyard | 94 | 62 | Oregon | Willamette Valley | Pinot Noir | Bergström | 27 | |
| US | Abetina | 94 | 105 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Ponzi | 28 |
| US | Garys’ Vineyard | 94 | 60 | California | Santa Lucia Highlands | Central Coast | Pinot Noir | Roar | 29 |
| US | The Funk Estate | 94 | 60 | Washington | Walla Walla Valley (WA) | Columbia Valley | Syrah | Saviah | 30 |
| Bulgaria | Bergulé | 90 | 15 | Bulgaria | Mavrud | Villa Melnik | 31 | ||
| US | Babushka | 90 | 37 | California | Russian River Valley | Sonoma | Chardonnay | Zepaltas | 32 |
| Italy | Vigna Piaggia | 90 | NA | Tuscany | Brunello di Montalcino | Sangiovese | Abbadia Ardenga | 33 | |
| France | Nonpareil Trésor Rosé Brut | 90 | 22 | France Other | Vin Mousseux | Sparkling Blend | Bouvet-Ladubay | 34 | |
| US | Conner Lee Vineyard | 90 | 42 | Washington | Columbia Valley (WA) | Columbia Valley | Chardonnay | Buty | 35 |
| Italy | Riserva | 90 | 135 | Tuscany | Brunello di Montalcino | Sangiovese | Carillon | 36 | |
| France | 90 | 60 | Rhône Valley | Châteauneuf-du-Pape | Rhône-style White Blend | Clos de L’Oratoire des Papes | 37 | ||
| Italy | 90 | 29 | Tuscany | Vino Nobile di Montepulciano | Sangiovese | Avignonesi | 38 | ||
| Italy | 90 | 23 | Tuscany | Chianti Classico | Sangiovese | Casina di Cornia | 39 | ||
| Italy | Riserva | 90 | 29 | Tuscany | Chianti Classico | Red Blend | Castello di Monterinaldi | 40 | |
| Spain | Amandi | 90 | 17 | Galicia | Ribeira Sacra | Mencía | Don Bernardino | 41 | |
| Spain | Alfonso Oloroso Seco | 90 | 26 | Andalucia | Jerez | Palomino | González Byass | 42 | |
| US | Private Reserve | 90 | 55 | Idaho | Petite Sirah | Huston | 43 | ||
| Italy | Riserva | 90 | 39 | Tuscany | Chianti Classico | Red Blend | Rignana | 44 | |
| France | Coteaux | 90 | 69 | Rhône Valley | Cornas | Syrah | Tardieu-Laurent | 45 | |
| Italy | Vigneto Odoardo Beccari Riserva | 90 | 30 | Tuscany | Chianti Classico | Red Blend | Vignavecchia | 46 | |
| Italy | Poggio alle Mura | 90 | 90 | Tuscany | Brunello di Montalcino | Sangiovese | Banfi | 47 | |
| US | Estate Grown | 90 | 60 | California | Mount Veeder | Napa | Cabernet Sauvignon | Brandlin | 48 |
| Italy | 90 | 50 | Tuscany | Brunello di Montalcino | Sangiovese | Brunelli Martoccia | 49 | ||
| US | 90 | 40 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | Canvasback | 50 | |
| Italy | Riserva | 90 | 100 | Tuscany | Brunello di Montalcino | Sangiovese | Capanne Ricci | 51 | |
| France | 90 | 68 | Burgundy | Chassagne-Montrachet | Chardonnay | Chartron et Trébuchet | 52 | ||
| France | Les 7 Hommes | 90 | 42 | Loire Valley | Sancerre | Sauvignon Blanc | Cherrier Frères | 53 | |
| France | L’Inédit | 90 | 28 | Loire Valley | Coteaux du Giennois | Pinot Noir | Clement et Florian Berthier | 54 | |
| US | 90 | 18 | California | Russian River Valley | Sonoma | Chardonnay | De Loach | 55 | |
| US | Four Flags | 90 | 69 | Washington | Red Mountain | Columbia Valley | Cabernet Sauvignon | DeLille | 56 |
| France | Le Pavé | 90 | NA | Loire Valley | Sancerre | Sauvignon Blanc | Domaine Vacheron | 57 | |
| US | Reserve | 90 | 25 | New York | Finger Lakes | Finger Lakes | Riesling | Dr. Konstantin Frank | 58 |
| US | Final Final | 90 | 30 | Washington | Columbia Valley (WA) | Columbia Valley | Cabernet Sauvignon-Syrah | Efeste | 59 |
| Italy | Poggio Bestiale | 90 | 60 | Tuscany | Maremma Toscana | Red Blend | Fattoria di Magliano | 60 | |
| Argentina | The Apple Doesn’t Fall Far From The Tree | 91 | 30 | Mendoza Province | Mendoza | Malbec | Matias Riccitelli | 61 | |
| Australia | 91 | 36 | Victoria | Mornington Peninsula | Pinot Noir | Moorooduc | 62 | ||
| Argentina | Alegoría Gran Reserva | 91 | 25 | Mendoza Province | Mendoza | Malbec | Navarro Correas | 63 | |
| France | L’Homme Mort Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Chenevières | 64 | |
| Portugal | 91 | 23 | Alentejano | Portuguese Red | Herdade do Rocim | 65 | |||
| US | Estate Select | 91 | 36 | California | Santa Clara Valley | Central Coast | Syrah | Jason-Stephens | 66 |
| France | Fourchaume Premier Cru | 91 | 38 | Burgundy | Chablis | Chardonnay | Louis Max | 67 | |
| US | Animo | 91 | 85 | California | Napa Valley | Napa | Cabernet Sauvignon | Michael Mondavi Family Estate | 68 |
| US | Schindler Vineyard | 91 | 50 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Panther Creek | 69 |
| US | Barrel Select | 91 | 60 | California | Rutherford | Napa | Cabernet Sauvignon | Provenance Vineyards | 70 |
| US | District Collection | 91 | 85 | California | St. Helena | Napa | Cabernet Sauvignon | Raymond | 71 |
| US | 91 | 45 | California | Sonoma Coast | Sonoma | Pinot Noir | Red Car | 72 | |
| Italy | Bussia Riserva | 91 | NA | Piedmont | Barolo | Nebbiolo | Silvano Bolmida | 73 | |
| US | 91 | 19 | Oregon | Willamette Valley | Willamette Valley | Pinot Gris | Trinity Vineyards | 74 | |
| Portugal | Premium | 91 | 15 | Alentejo | Portuguese Red | Adega Cooperativa de Borba | 75 | ||
| US | Premier Cuvée | 91 | 54 | Oregon | Willamette Valley | Willamette Valley | Pinot Noir | Archery Summit | 76 |
| France | Le Nombre d’Or Brut Nature | 91 | 85 | Champagne | Champagne | Chardonnay | Aubry | 77 | |
| US | Juliana Vineyard | 91 | 38 | California | Napa Valley | Napa | Sauvignon Blanc | B Cellars | 78 |
| US | 91 | 28 | California | Napa Valley | Napa | Cabernet Sauvignon | B Side | 79 | |
| Italy | Boscato | 91 | 75 | Piedmont | Barolo | Nebbiolo | Bel Colle | 80 | |
| US | Aeolian | 91 | 42 | Oregon | Eola-Amity Hills | Willamette Valley | Pinot Noir | Bethel Heights | 81 |
| Israel | Reserve | 91 | 25 | Upper Galilee | Cabernet Sauvignon | Binyamina | 82 | ||
| Italy | Palliano Riserva | 91 | NA | Piedmont | Roero | Nebbiolo | Ceste | 83 | |
| Italy | del Comune di Serralunga d’Alba | 91 | 59 | Piedmont | Barolo | Nebbiolo | Cascina Cucco | 84 | |
| Italy | Bricco Luciani | 91 | 85 | Piedmont | Barolo | Nebbiolo | Cascina del Monastero | 85 | |
| Italy | Bricco Gattera | 91 | 80 | Piedmont | Barolo | Nebbiolo | Cordero di Montezemolo | 86 | |
| France | Montmains Premier Cru | 91 | 45 | Burgundy | Chablis | Chardonnay | Domaine Gérard Duplessis | 87 | |
| US | 91 | 22 | California | Napa Valley | Napa | Cabernet Sauvignon | Eagle Glen | 88 | |
| US | Bacigalupi Vineyard | 91 | 65 | California | Russian River Valley | Sonoma | Pinot Noir | Eleven Eleven | 89 |
| US | Magnificat | 91 | 50 | California | Napa Valley | Napa | Meritage | Franciscan | 90 |
| US | 86 | 10 | California | California | California Other | Cabernet Sauvignon | Belle Ambiance | 91 | |
| Portugal | Marquês de Marialva Rosé Bruto | 86 | 12 | Beira Atlantico | Baga | Adega de Cantanhede | 92 | ||
| Italy | Nature | 86 | 22 | Veneto | Prosecco | Glera | De Stefani | 93 | |
| US | Small Lot Blend | 86 | 13 | California | Mendocino County | Mendocino/Lake Counties | Chardonnay | Parducci | 94 |
| Portugal | Muros de Vinha | 86 | 10 | Douro | Portuguese Red | Quinta do Portal | 95 | ||
| France | Château Beauvillain-Monpezat | 86 | 14 | Southwest France | Cahors | Malbec-Merlot | Rigal | 96 | |
| US | 86 | 18 | California | California | California Other | Chardonnay | The Naked Grape | 97 | |
| US | 86 | 36 | California | Sonoma Valley | Sonoma | Cabernet Sauvignon | Tin Barn | 98 | |
| France | Pigmentum | 86 | 15 | Southwest France | Buzet | Merlot-Malbec | Georges Vigouroux | 99 | |
| France | Pigmentum | 86 | 10 | Southwest France | Côtes de Gascogne | Ugni Blanc-Colombard | Georges Vigouroux | 100 |
| price | points |
|---|---|
| 235 | 96 |
| 110 | 96 |
| 90 | 96 |
| 65 | 96 |
| 66 | 95 |
| 73 | 95 |
| 65 | 95 |
| 110 | 95 |
| 65 | 95 |
| 60 | 95 |
| 80 | 95 |
| 48 | 95 |
| 48 | 95 |
| 90 | 95 |
| 185 | 95 |
| 90 | 95 |
| 325 | 95 |
| 80 | 95 |
| 290 | 95 |
| 75 | 95 |
| 24 | 95 |
| 79 | 95 |
| 220 | 95 |
| 60 | 95 |
| 45 | 95 |
| 57 | 94 |
| 62 | 94 |
| 105 | 94 |
| 60 | 94 |
| 60 | 94 |
| 15 | 90 |
| 37 | 90 |
| NA | 90 |
| 22 | 90 |
| 42 | 90 |
| 135 | 90 |
| 60 | 90 |
| 29 | 90 |
| 23 | 90 |
| 29 | 90 |
| 17 | 90 |
| 26 | 90 |
| 55 | 90 |
| 39 | 90 |
| 69 | 90 |
| 30 | 90 |
| 90 | 90 |
| 60 | 90 |
| 50 | 90 |
| 40 | 90 |
| 100 | 90 |
| 68 | 90 |
| 42 | 90 |
| 28 | 90 |
| 18 | 90 |
| 69 | 90 |
| NA | 90 |
| 25 | 90 |
| 30 | 90 |
| 60 | 90 |
| 30 | 91 |
| 36 | 91 |
| 25 | 91 |
| 45 | 91 |
| 23 | 91 |
| 36 | 91 |
| 38 | 91 |
| 85 | 91 |
| 50 | 91 |
| 60 | 91 |
| 85 | 91 |
| 45 | 91 |
| NA | 91 |
| 19 | 91 |
| 15 | 91 |
| 54 | 91 |
| 85 | 91 |
| 38 | 91 |
| 28 | 91 |
| 75 | 91 |
| 42 | 91 |
| 25 | 91 |
| NA | 91 |
| 59 | 91 |
| 85 | 91 |
| 80 | 91 |
| 45 | 91 |
| 22 | 91 |
| 65 | 91 |
| 50 | 91 |
| 10 | 86 |
| 12 | 86 |
| 22 | 86 |
| 13 | 86 |
| 10 | 86 |
| 14 | 86 |
| 18 | 86 |
| 36 | 86 |
| 15 | 86 |
| 10 | 86 |
idea:
\[\begin{equation} y_i = \alpha + \beta x_i + \epsilon_i \end{equation}\]
where \(\epsilon_i\) is an error term. Define mean-squared loss:
\[\begin{equation} L = \frac{1}{K} \sum_{i=1}^{K} (y_i - (\alpha + \beta x_i))^2 \end{equation}\]
determine \(\hat{\alpha}\) and \(\hat{\beta}\) as those minimising \(L\):
\[\begin{align} \frac{\partial L}{\partial \alpha} &= -\frac{2}{K}\sum_{i=1}^{K} (y_i - (\alpha + \beta x_i)) = 0\\ \frac{\partial L}{\partial \beta} &= -\frac{2}{K}\sum_{i=1}^{K} x_i (y_i - (\alpha + \beta x_i)) = 0 \end{align}\]
although a closed form expression exists for \(\hat{\alpha}\) and \(\hat{\beta}\), for more general models, one doesn’t exist \(\implies\) use gradient descent optimisation
\[\begin{align} \alpha &= \alpha - \eta \frac{\partial L}{\partial \alpha}\\ \beta &= \beta - \eta \frac{\partial L}{\partial \beta} \end{align}\]
until \(\alpha\) and \(\beta\) no longer change. \(\eta\) is the learning rate
\[\begin{equation} y_i = \theta_0 + \theta_1 x_i + \theta_2 x_i^2 + ... + \theta_p x_i^p + \epsilon_i \end{equation}\]
model is better able to fit more complex datasets
\[\begin{equation} L = C||\theta||_q + \frac{1}{K} \sum_{i=1}^{K} (y_i - f_p(x_i))^2 \end{equation}\]
where \(||.||_q\) denotes the \(L_q\) norm: different choices can yield very different estimates
for new data point \(\tilde x_i\):
for new data point \(\tilde x_i\):
many options possible. Common metrics include:
\[\begin{equation} s(x_1,x_2) = \frac{x_1.x_2}{|x_1||x_2|} \end{equation}\]
assume
\[\begin{equation} \boldsymbol{x} \sim \mathcal{N}(0, I) \end{equation}\]
where \(I\in\mathbb{R}^d\). What does the distribution of Euclidean distances between points look like as \(d\) changes?
\[\begin{equation} y_i \sim \text{Bernoulli}(\theta_i) \end{equation}\]
where \(0\leq \theta_i \leq 1 = Pr(y_i=1)\)
is given by:
\[\begin{equation} \text{Pr}(y_i|\theta_i) = \theta_i^{y_i} (1 - \theta_i)^{1 - y_i} \end{equation}\]
so that \(\text{Pr}(y_i=1) = \theta_i\) and \(\text{Pr}(y_i=0) = 1 - \theta_i\)
In logistic regression, we use logistic function:
\[\begin{equation} \theta_i = f_\beta(x_i) := \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_i))} \end{equation}\]
assume data are i.i.d., the likelihood is:
\[\begin{equation} L=p(\boldsymbol{y}|\beta,\boldsymbol{x}) = \prod_{i=1}^{K} f_\beta(x_i)^{y_i} (1 - f_\beta(x_i))^{1 - y_i}. \end{equation}\]
Can use gradient descent to find maximum likelihood estimates (or estimate using Bayesian inference).
straightforward to extend the model to incorporate multiple regressions:
\[\begin{equation} f_\beta(x_i) := \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i}))} \end{equation}\]
But how to interpret parameters of logistic regression?
another way of writing logistic function:
\[\begin{align} f_\beta(x_i) &= \frac{1}{1 + \exp (-(\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i}))}\\ &= \frac{\exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i})}{1 + \exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i})} \end{align}\]
so that
\[\begin{align} 1 - f_\beta(x_i) = \frac{1}{1 + \exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i})} \end{align}\]
taking the ratio:
\[\begin{equation} \text{odds} = \frac{f_\beta(x_i)}{1-f_\beta(x_i)} = \exp (\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i}) \end{equation}\]
so that
\[\begin{equation} \log\text{odds} =\beta_0 + \beta_1 x_{1,i} + ... + \beta_p x_{p,i} \end{equation}\]
meaning (say) \(\beta_1\) represents the change in log-odds for a one unit change in \(x_{1}\)
All available on SOLO:
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